\(\int \frac {(c+d x+e x^2) (a+b x^3)^3}{x} \, dx\) [328]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=a^3 d x+\frac {1}{2} a^3 e x^2+a^2 b c x^3+\frac {3}{4} a^2 b d x^4+\frac {3}{5} a^2 b e x^5+\frac {1}{2} a b^2 c x^6+\frac {3}{7} a b^2 d x^7+\frac {3}{8} a b^2 e x^8+\frac {1}{9} b^3 c x^9+\frac {1}{10} b^3 d x^{10}+\frac {1}{11} b^3 e x^{11}+a^3 c \log (x) \]

[Out]

a^3*d*x+1/2*a^3*e*x^2+a^2*b*c*x^3+3/4*a^2*b*d*x^4+3/5*a^2*b*e*x^5+1/2*a*b^2*c*x^6+3/7*a*b^2*d*x^7+3/8*a*b^2*e*
x^8+1/9*b^3*c*x^9+1/10*b^3*d*x^10+1/11*b^3*e*x^11+a^3*c*ln(x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1642} \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=a^3 c \log (x)+a^3 d x+\frac {1}{2} a^3 e x^2+a^2 b c x^3+\frac {3}{4} a^2 b d x^4+\frac {3}{5} a^2 b e x^5+\frac {1}{2} a b^2 c x^6+\frac {3}{7} a b^2 d x^7+\frac {3}{8} a b^2 e x^8+\frac {1}{9} b^3 c x^9+\frac {1}{10} b^3 d x^{10}+\frac {1}{11} b^3 e x^{11} \]

[In]

Int[((c + d*x + e*x^2)*(a + b*x^3)^3)/x,x]

[Out]

a^3*d*x + (a^3*e*x^2)/2 + a^2*b*c*x^3 + (3*a^2*b*d*x^4)/4 + (3*a^2*b*e*x^5)/5 + (a*b^2*c*x^6)/2 + (3*a*b^2*d*x
^7)/7 + (3*a*b^2*e*x^8)/8 + (b^3*c*x^9)/9 + (b^3*d*x^10)/10 + (b^3*e*x^11)/11 + a^3*c*Log[x]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 d+\frac {a^3 c}{x}+a^3 e x+3 a^2 b c x^2+3 a^2 b d x^3+3 a^2 b e x^4+3 a b^2 c x^5+3 a b^2 d x^6+3 a b^2 e x^7+b^3 c x^8+b^3 d x^9+b^3 e x^{10}\right ) \, dx \\ & = a^3 d x+\frac {1}{2} a^3 e x^2+a^2 b c x^3+\frac {3}{4} a^2 b d x^4+\frac {3}{5} a^2 b e x^5+\frac {1}{2} a b^2 c x^6+\frac {3}{7} a b^2 d x^7+\frac {3}{8} a b^2 e x^8+\frac {1}{9} b^3 c x^9+\frac {1}{10} b^3 d x^{10}+\frac {1}{11} b^3 e x^{11}+a^3 c \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=a^3 d x+\frac {1}{2} a^3 e x^2+a^2 b c x^3+\frac {3}{4} a^2 b d x^4+\frac {3}{5} a^2 b e x^5+\frac {1}{2} a b^2 c x^6+\frac {3}{7} a b^2 d x^7+\frac {3}{8} a b^2 e x^8+\frac {1}{9} b^3 c x^9+\frac {1}{10} b^3 d x^{10}+\frac {1}{11} b^3 e x^{11}+a^3 c \log (x) \]

[In]

Integrate[((c + d*x + e*x^2)*(a + b*x^3)^3)/x,x]

[Out]

a^3*d*x + (a^3*e*x^2)/2 + a^2*b*c*x^3 + (3*a^2*b*d*x^4)/4 + (3*a^2*b*e*x^5)/5 + (a*b^2*c*x^6)/2 + (3*a*b^2*d*x
^7)/7 + (3*a*b^2*e*x^8)/8 + (b^3*c*x^9)/9 + (b^3*d*x^10)/10 + (b^3*e*x^11)/11 + a^3*c*Log[x]

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87

method	result	size
default	\(a^{3} d x +\frac {a^{3} e \,x^{2}}{2}+a^{2} x^{3} b c +\frac {3 a^{2} b d \,x^{4}}{4}+\frac {3 a^{2} b e \,x^{5}}{5}+\frac {a \,b^{2} c \,x^{6}}{2}+\frac {3 a \,b^{2} d \,x^{7}}{7}+\frac {3 a \,b^{2} e \,x^{8}}{8}+\frac {b^{3} c \,x^{9}}{9}+\frac {b^{3} d \,x^{10}}{10}+\frac {b^{3} e \,x^{11}}{11}+a^{3} c \ln \left (x \right )\)	\(110\)
norman	\(a^{3} d x +\frac {a^{3} e \,x^{2}}{2}+a^{2} x^{3} b c +\frac {3 a^{2} b d \,x^{4}}{4}+\frac {3 a^{2} b e \,x^{5}}{5}+\frac {a \,b^{2} c \,x^{6}}{2}+\frac {3 a \,b^{2} d \,x^{7}}{7}+\frac {3 a \,b^{2} e \,x^{8}}{8}+\frac {b^{3} c \,x^{9}}{9}+\frac {b^{3} d \,x^{10}}{10}+\frac {b^{3} e \,x^{11}}{11}+a^{3} c \ln \left (x \right )\)	\(110\)
risch	\(a^{3} d x +\frac {a^{3} e \,x^{2}}{2}+a^{2} x^{3} b c +\frac {3 a^{2} b d \,x^{4}}{4}+\frac {3 a^{2} b e \,x^{5}}{5}+\frac {a \,b^{2} c \,x^{6}}{2}+\frac {3 a \,b^{2} d \,x^{7}}{7}+\frac {3 a \,b^{2} e \,x^{8}}{8}+\frac {b^{3} c \,x^{9}}{9}+\frac {b^{3} d \,x^{10}}{10}+\frac {b^{3} e \,x^{11}}{11}+a^{3} c \ln \left (x \right )\)	\(110\)
parallelrisch	\(a^{3} d x +\frac {a^{3} e \,x^{2}}{2}+a^{2} x^{3} b c +\frac {3 a^{2} b d \,x^{4}}{4}+\frac {3 a^{2} b e \,x^{5}}{5}+\frac {a \,b^{2} c \,x^{6}}{2}+\frac {3 a \,b^{2} d \,x^{7}}{7}+\frac {3 a \,b^{2} e \,x^{8}}{8}+\frac {b^{3} c \,x^{9}}{9}+\frac {b^{3} d \,x^{10}}{10}+\frac {b^{3} e \,x^{11}}{11}+a^{3} c \ln \left (x \right )\)	\(110\)

[In]

int((e*x^2+d*x+c)*(b*x^3+a)^3/x,x,method=_RETURNVERBOSE)

[Out]

a^3*d*x+1/2*a^3*e*x^2+a^2*x^3*b*c+3/4*a^2*b*d*x^4+3/5*a^2*b*e*x^5+1/2*a*b^2*c*x^6+3/7*a*b^2*d*x^7+3/8*a*b^2*e*
x^8+1/9*b^3*c*x^9+1/10*b^3*d*x^10+1/11*b^3*e*x^11+a^3*c*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=\frac {1}{11} \, b^{3} e x^{11} + \frac {1}{10} \, b^{3} d x^{10} + \frac {1}{9} \, b^{3} c x^{9} + \frac {3}{8} \, a b^{2} e x^{8} + \frac {3}{7} \, a b^{2} d x^{7} + \frac {1}{2} \, a b^{2} c x^{6} + \frac {3}{5} \, a^{2} b e x^{5} + \frac {3}{4} \, a^{2} b d x^{4} + a^{2} b c x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x + a^{3} c \log \left (x\right ) \]

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a)^3/x,x, algorithm="fricas")

[Out]

1/11*b^3*e*x^11 + 1/10*b^3*d*x^10 + 1/9*b^3*c*x^9 + 3/8*a*b^2*e*x^8 + 3/7*a*b^2*d*x^7 + 1/2*a*b^2*c*x^6 + 3/5*
a^2*b*e*x^5 + 3/4*a^2*b*d*x^4 + a^2*b*c*x^3 + 1/2*a^3*e*x^2 + a^3*d*x + a^3*c*log(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=a^{3} c \log {\left (x \right )} + a^{3} d x + \frac {a^{3} e x^{2}}{2} + a^{2} b c x^{3} + \frac {3 a^{2} b d x^{4}}{4} + \frac {3 a^{2} b e x^{5}}{5} + \frac {a b^{2} c x^{6}}{2} + \frac {3 a b^{2} d x^{7}}{7} + \frac {3 a b^{2} e x^{8}}{8} + \frac {b^{3} c x^{9}}{9} + \frac {b^{3} d x^{10}}{10} + \frac {b^{3} e x^{11}}{11} \]

[In]

integrate((e*x**2+d*x+c)*(b*x**3+a)**3/x,x)

[Out]

a**3*c*log(x) + a**3*d*x + a**3*e*x**2/2 + a**2*b*c*x**3 + 3*a**2*b*d*x**4/4 + 3*a**2*b*e*x**5/5 + a*b**2*c*x*
*6/2 + 3*a*b**2*d*x**7/7 + 3*a*b**2*e*x**8/8 + b**3*c*x**9/9 + b**3*d*x**10/10 + b**3*e*x**11/11

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=\frac {1}{11} \, b^{3} e x^{11} + \frac {1}{10} \, b^{3} d x^{10} + \frac {1}{9} \, b^{3} c x^{9} + \frac {3}{8} \, a b^{2} e x^{8} + \frac {3}{7} \, a b^{2} d x^{7} + \frac {1}{2} \, a b^{2} c x^{6} + \frac {3}{5} \, a^{2} b e x^{5} + \frac {3}{4} \, a^{2} b d x^{4} + a^{2} b c x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x + a^{3} c \log \left (x\right ) \]

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a)^3/x,x, algorithm="maxima")

[Out]

1/11*b^3*e*x^11 + 1/10*b^3*d*x^10 + 1/9*b^3*c*x^9 + 3/8*a*b^2*e*x^8 + 3/7*a*b^2*d*x^7 + 1/2*a*b^2*c*x^6 + 3/5*
a^2*b*e*x^5 + 3/4*a^2*b*d*x^4 + a^2*b*c*x^3 + 1/2*a^3*e*x^2 + a^3*d*x + a^3*c*log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=\frac {1}{11} \, b^{3} e x^{11} + \frac {1}{10} \, b^{3} d x^{10} + \frac {1}{9} \, b^{3} c x^{9} + \frac {3}{8} \, a b^{2} e x^{8} + \frac {3}{7} \, a b^{2} d x^{7} + \frac {1}{2} \, a b^{2} c x^{6} + \frac {3}{5} \, a^{2} b e x^{5} + \frac {3}{4} \, a^{2} b d x^{4} + a^{2} b c x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x + a^{3} c \log \left ({\left | x \right |}\right ) \]

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a)^3/x,x, algorithm="giac")

[Out]

1/11*b^3*e*x^11 + 1/10*b^3*d*x^10 + 1/9*b^3*c*x^9 + 3/8*a*b^2*e*x^8 + 3/7*a*b^2*d*x^7 + 1/2*a*b^2*c*x^6 + 3/5*
a^2*b*e*x^5 + 3/4*a^2*b*d*x^4 + a^2*b*c*x^3 + 1/2*a^3*e*x^2 + a^3*d*x + a^3*c*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x} \, dx=\frac {b^3\,c\,x^9}{9}+\frac {a^3\,e\,x^2}{2}+\frac {b^3\,d\,x^{10}}{10}+\frac {b^3\,e\,x^{11}}{11}+a^3\,c\,\ln \left (x\right )+a^3\,d\,x+a^2\,b\,c\,x^3+\frac {a\,b^2\,c\,x^6}{2}+\frac {3\,a^2\,b\,d\,x^4}{4}+\frac {3\,a\,b^2\,d\,x^7}{7}+\frac {3\,a^2\,b\,e\,x^5}{5}+\frac {3\,a\,b^2\,e\,x^8}{8} \]

[In]

int(((a + b*x^3)^3*(c + d*x + e*x^2))/x,x)

[Out]

(b^3*c*x^9)/9 + (a^3*e*x^2)/2 + (b^3*d*x^10)/10 + (b^3*e*x^11)/11 + a^3*c*log(x) + a^3*d*x + a^2*b*c*x^3 + (a*
b^2*c*x^6)/2 + (3*a^2*b*d*x^4)/4 + (3*a*b^2*d*x^7)/7 + (3*a^2*b*e*x^5)/5 + (3*a*b^2*e*x^8)/8